In a direct proof that 'if n is even, then n² is even,' a student writes: 'Let n = 2k. Then n² = 4k² = 2(2k²), which has the form 2m and is therefore even.' Which step is the crucial move that makes this a proof?
ASquaring n to get 4k², since this introduces the algebraic structure needed
BWriting n = 2k, because this unpacks the hypothesis into usable algebraic form
CNoting that 4k² = 2(2k²), since this is where the conclusion is established
DConcluding that n² is even, since this names the result
The key move is unpacking the hypothesis: writing 'n is even' as 'n = 2k for some integer k.' This transforms an abstract property into a concrete algebraic object that can be manipulated. Everything that follows — squaring, factoring, recognizing the form 2m — is routine algebra. Direct proof works by finding the algebraic machinery hidden inside the hypothesis; that unpacking step is where the proof actually begins.
Question 2 Multiple Choice
A student attempting to prove 'if n² is even, then n is even' writes: 'Assume n² is even. Since n² is even, n must be even, so n is even.' What logical error did the student commit?
AAssuming the hypothesis — the student should not assume anything in a direct proof
BBegging the question — the student used the conclusion ('n is even') as a step in the argument
CCircular reasoning — the student repeated the hypothesis instead of using it
DA valid proof — the step 'n must be even' follows directly from 'n² is even'
The step 'since n² is even, n must be even' is precisely the conclusion the student is supposed to prove — not a logical step that follows from anything established so far. This is begging the question (petitio principii): assuming the conclusion as part of the argument for the conclusion. Assuming the hypothesis (H) is correct and not circular; the error is assuming P (the conclusion) in order to prove P. Notably, this particular theorem is hard to prove by direct method — contrapositive ('if n is odd, then n² is odd') works more cleanly.
Question 3 True / False
In a direct proof of 'if H then P,' assuming H at the start of the proof is a form of circular reasoning.
TTrue
FFalse
Answer: False
Assuming H is exactly what the conditional statement licenses you to do — you are not claiming H is universally true, but showing what follows under the assumption that H holds. Circular reasoning would be using P (the conclusion) somewhere in the argument to establish P. Assuming the hypothesis is the starting point of every direct proof; using the conclusion is the error. The distinction is between 'assume H to prove P' (correct) and 'assume P to prove P' (circular).
Question 4 True / False
Direct proof is the most effective method for nearly every conditional statement of the form 'if H then P.'
TTrue
FFalse
Answer: False
Direct proof works well when the hypothesis, when unpacked, contains the algebraic or logical machinery needed to produce the conclusion. When the hypothesis and conclusion feel 'far apart' — when reasoning forward from H doesn't naturally lead to P — an indirect method (proof by contrapositive or proof by contradiction) is usually cleaner. For example, 'if n² is even then n is even' is awkward by direct proof but elegant by contrapositive. Choosing the right proof strategy is itself a mathematical skill.
Question 5 Short Answer
What is the difference between 'assuming the hypothesis' and 'begging the question,' and why does only one of them make a direct proof invalid?
Think about your answer, then reveal below.
Model answer: Assuming the hypothesis means taking H as given at the start and reasoning forward to show P follows — this is exactly what proving 'if H then P' requires. Begging the question means using P (the conclusion) as a step in the argument for P — sneaking in what you're trying to prove as if it were already established. Only the second is logically invalid: a proof of 'if H then P' is supposed to show that P follows from H, so using P before it's been established makes the argument circular. Assuming H doesn't make the proof circular because H and P are different statements.
The key is which statement you're assuming. You are licensed to assume H because that's the hypothesis of the conditional you're proving — you're not proving H is always true. But you haven't established P yet, so you cannot use P as a reason for P. Direct proof is essentially an exercise in following the logical implications of H step by step until P emerges — any step that relies on P before that emergence is circular.